We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness, and are stable in some cardinal. Assuming the singular cardinal hypothesis, we prove a full characterization of the stability cardinals, and connect the stability spectrum with the behavior of saturated models.We deduce that if a class is stable on a tail of cardinals, then it has no long splitting chains. This indicates that there is a clear notion of superstability in this framework.We also present an (...) application to homogeneous model theory: for [Formula: see text] a homogeneous diagram in a first-order theory [Formula: see text], if [Formula: see text] is both stable in [Formula: see text] and categorical in [Formula: see text] then [Formula: see text] is stable in all [Formula: see text]. (shrink)

We show how to build prime models in classes of saturated models of abstract elementary classes (AECs) having a well‐behaved independence relation: Let be an almost fully good AEC that is categorical in and has the ‐existence property for domination triples. For any, the class of Galois saturated models of of size λ has prime models over every set of the form. This generalizes an argument of Shelah, who proved the result when λ is a successor cardinal.

Theorem 0.1LetK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}be an abstract elementary class with amalgamation and no maximal models. Letλ>LS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda > {LS}$$\end{document}. IfK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}is categorical inλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, then the model of cardinalityλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}is Galois-saturated.This answers a question asked independently by Baldwin and (...) Shelah. We deduce several corollaries: K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document} has a unique limit model in each cardinal below λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document} is weakly tame below λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, and the thresholds of several existing categoricity transfers can be improved.We also prove a downward transfer of solvability :Corollary 0.2LetK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}be an AEC with amalgamation and no maximal models. Letλ>μ>LS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda> \mu > {LS}$$\end{document}. IfK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}is solvable inλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, thenK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}is solvable inμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}. (shrink)

In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for μ‐forking, a certain independence notion arising from splitting. This had been a longstanding technical difficulty when constructing forking‐like notions in this setup. As an application, we show that the two versions of forking symmetry appearing in the literature (the one defined by Shelah for good frames and the one defined by VanDieren for splitting) are equivalent.

Assuming the existence of a monster model, tameness, and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>\operatorname {LS}(\mathbf {K})$ be a regular stability cardinal and let $\chi $ be the local character of $\mu $ -nonsplitting. The following holds: 1.When $\mu $ -nonforking is restricted to $(\mu,\geq \chi )$ -limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension, and continuity. It also has local character $\chi $. This generalizes (...) Vasey’s result [37, Corollary 13.16] which assumed $\mu $ -superstability to obtain same properties but with local character $\aleph _0$.2.There is $\lambda \in [\mu,h(\mu ))$ such that if $\mathbf {K}$ is stable in every cardinal between $\mu $ and $\lambda $, then $\mathbf {K}$ has $\mu $ -symmetry while $\mu $ -nonforking in (1) has symmetry. In this case:(a) $\mathbf {K}$ has the uniqueness of $(\mu,\geq \chi )$ -limit models: if $M_1,M_2$ are both $(\mu,\geq \chi )$ -limit over some $M_0\in K_{\mu }$, then $M_1\cong _{M_0}M_2$ ;(b)any increasing chain of $\mu ^+$ -saturated models of length $\geq \chi $ has a $\mu ^+$ -saturated union. These generalize [31] and remove the symmetry assumption in [10, 38].Under $(<\mu )$ -tameness, the conclusions of (1), (2)(a)(b) are equivalent to $\mathbf {K}$ having the $\chi $ -local character of $\mu $ -nonsplitting.Grossberg and Vasey [18, 38] gave eventual superstability criteria for tame AECs with a monster model. We remove the high cardinal threshold and reduce the cardinal jump between equivalent superstability criteria. We also add two new superstability criteria to the list: a weaker version of solvability and the boundedness of the U-rank. (shrink)

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes with amalgamation. We prove:Theorem 0.1.IfKis a tame AEC with amalgamation satisfying a natural definition of superstability, then for all high-enoughλ:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda {:}$$\end{document}The union of an increasing chain ofλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-saturated models isλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-saturated.There exists a type-full goodλ\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-frame with underlying class the saturated models of sizeλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}.There exists a unique limit model of sizeλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}. Our proofs use independence calculus and a generalization of averages to this non first-order context. (shrink)

For a fixed natural number \, the Hart–Shelah example is an abstract elementary class with amalgamation that is categorical exactly in the infinite cardinals less than or equal to \. We investigate recently-isolated properties of AECs in the setting of this example. We isolate the exact amount of type-shortness holding in the example and show that it has a type-full good \-frame which fails the existence property for uniqueness triples. This gives the first example of such a frame. Along the (...) way, we develop new tools to build and analyze good frames. (shrink)

In this paper, we study a stability transfer theorem in d-tame metric abstract elementary classes, in a similar way as in 2, but using superstability-like assumptions which involves a new independence notion instead of ℵ0-locality.